data structures for 3d meshes - cnrvcg.isti.cnr.it/~cignoni/fgt1516/fgt_02_data structures.pdfdata...
TRANSCRIPT
2
Surfaces
A 2-d im ens iona l reg ion o f 3D spaceA portion o f space hav ing leng th and
breadth but no th ickness
3
Defining Surfaces
Analytica lly...Pa ram e tr ic su rfaces
Im plic it surfaces
4
Representing Real World Surfaces
Analytic definition falls short of representing real world surfaces in a tractable way
... surfaces can be represented by cell complexes
5
Cell complexes (meshes)
Intu itive description: a continuous surface d iv ided in po lygons
quadrilaterals (quads)
triangles Generic polygons
6
Ce ll Com p lexes (m eshes)In nature, meshes arise in a variety of
contexts:Cells in organic tissuesCrystalsMolecules…Mostly convex but irregular cellsCommon concept: complex shapes can be
described as collections of simple building blocks
6
7
Cell Complexes (meshes)
m ore form al defin ition a ce ll is a convex po lytope in a proper face of a ce ll is a convex po lytope
in
8
Cell Complexes (meshes)
a co llection of ce lls is a com plex iff every face of a ce ll be longs to the com plex For every ce lls C and C ’, the ir in tersection
e ither is em pty or is a com m on face of both
9
Maximal Cell Complex
the order of a cell is the number of its sides (or vertices)
a complex is a k-complex if the maximum of the order of its cells is k
a cell is maximal if it is not a face of another cell a k-complex is maximal iff all maximal cells have
order k short form : no dangling edges!
10
Simplicial Complex
A ce ll com plex is a sim plicial com plex when the cells are sim plexes
A d-s im p lex is the convex hu ll o f d+1 po ints in
0-simplex 1-simplex 2-simplex 3-simplex
11
Meshes, at last
When talking of triangle mesh the intended meaning is a maximal 2-simplicial complex
Topology vs Geometry
Di un complesso simpliciale e' buona norma distinguereRealizzazione geometrica
Dove stanno effettivamente nello spazio i vertici del nostro complesso simpliciale
Caratterizzazione topologicaCome sono connessi combinatoriamente i vari
elementi
Topology vs geometry 2
Nota: Di uno stesso oggetto e' possibile dare rappresentazioni con eguale realizzazione geometrica ma differente caratterizzazione topologica (molto differente!) Demo kleine
Nota: Di un oggetto si puo' dire molte cose considerandone solo la componente topologicaOrientabilitacomponenti connessebordi
14
Manifoldness
a surface S is 2-manifold iff:the neighborhood of each point is
homeomorphic to Euclidean space in two dimensionor … in other words..
the neighborhood of each point is homeomorphic to a disk (or a semidisk if the surface has boundary)
15
Orientability
A surface is orientable if it is possible to make a consistent choice for the normal vector …it has two sides
Moebius strips, klein bottles, and non manifold surfaces are not orientable
Incidenza Adiacenza
Due simplessi σ e σ' sono incidenti se σ è una faccia propria di σ' o vale il viceversa.
Due k-simplessi sono m-adiacenti (k>m) se esiste un m-simplesso che è una faccia propria di entrambi.Due triangoli che condividono
un edge sono 1 -adiacentiDue triangoli che condividono
un vertice sono 0-adiacenti
Relazioni di Adiacenza
Per semplicità nel caso di mesh si una relazione di adiacenza con un una coppia (ordinata!) di lettere che indicano le entità coinvolteFF adiacenza tra triangoliFV i vertici che compongono un triangoloVF i triangoli incidenti su un dato vertice
Relazioni di adiacenza
Di tutte le possibili relazioni di adiacenza di solito vale la pena se ne considera solo un sottoinsieme (su 9) e ricavare le altre proceduralmente
Relazioni di adiacenza FF ~ 1-adiacenza
EE ~ 0 adiacenza
FE ~ sottofacce proprie di F con dim 1
FV ~ sottofacce proprie di F con dim 0
EV ~ sottofacce proprie di E con dim 0
VF ~ F in Σ : V sub faccia di F
VE ~ E in Σ : V sub faccia di E
EF ~ F in Σ : E sub faccia di F
VV ~ V' in Σ : Esiste E (V,V')
Partial adiacency
Per risparmiare a volte si mantiene una informazione di adiacenza parziale VF* memorizzo solo un riferimento dal
vertice ad una delle facce e poi 'navigo' sulla mesh usando la FF per trovare le altre facce incidenti su V
Relazioni di adiacenza
In un 2-complesso simpliciale immerso in R3, che sia 2 manifoldFV FE FF EF EV sono di cardinalità bounded
(costante nel caso non abbia bordi)|FV|= 3 |EV| = 2 |FE| = 3 |FF| <= 2 |EF| <= 2
VV VE VF EE sono di card. variabile ma in stimabile in media|VV|~|VE|~|VF|~6|EE|~10F ~ 2V
22
Genus
The Genus of a closed surface, orientable and 2-manifold is the maximum number of cuts we can make along non intersecting closed curves without splitting the surface in two.
…also known as the number of handles
0 1 2
23
Euler characteristic
24
Euler characteristics
= 2 for any simply connected polyhedron proof by construction… play with examples:
χ
25
Euler characteristics
let’s try a more complex figure…
why =0 ?
26
Euler characteristics
where g is the genus of the surface
χ=2−2g
27
Euler characteristics
let’s try a more complex figure…remove a face. The surface is not closed anymore
why =-1 ?
28
Euler characteristics
where b is the num ber of borders of the surface
χ=2−2g−b
29
Euler characteristics
Remove the border by adding a new vertex and connecting all the k vertices on the border to it.
A A’
X' = X + V' -E' + F' = X + 1 – k + k = X +1
Differential quantities:normals
The (unit) normal to a point is the (unit) vector perpendicular to the tangent plane
implicit surface : f ( x , y , z)=0
n= ∇ f[∇ f ]
parametric surface :f (u ,v)=( f x (u , v) , f y (u , v) , f z (u , v))
n= ∂∂u
f × ∂∂ v
f
Normals on triangle meshes
Computed per-vertex and interpolated over the faces
Common: consider the tangent plane as the average among the planes containing all the faces incident on the vertex
Normals on triangle meshes
Does it work? Yes, for a “good” tessellationSmall triangles may change the result
dramatically Weighting by edge length / area / angle
helps
Differential quantities: Curvature
The curvature is a measure of how much a line is curve
x (t) , y (t) a parametric curveϕ tangential angle%s arc lenght
κ≡d ϕds
=d ϕ/dtds /dt
=d ϕ/dt
√( dxdt
)2
+( dydt
)2=
d ϕ/dt
√ x ' 2+ y ' 2
κ= x ' y ' '− y ' x ' '( x ' 2+ y ' 2)3/2
Curvature on a surface
Given the normal at point p and a tangent direction
The curvature along is the 2D curvature of the intersection between the plane and the surface
θθ
Curvatures
A curvature for each direction Take the two directions for which curvature is max and
min
the directions of max and min curvature are orthogonal
κ1 ,κ2 principal curvaturese1 , e2 principal directions
[Meyer02]
Gaussian and Mean curvature
Gaussian curvature: the product of principal curvatures
Mean curvature: the average of principal curvatures
κG≡Κ≡κ1⋅κ2
κ̄≡H≡κ1+κ2
2
Examples..
Red:low → red:high (not in the same scale)
mean gaussian min max
[Meyer02]
Gaussian Curvature
Gaussian curvature is an intrinsic propertyIt can be computed by a bidimensional
inhabitant of the surface by walking around a fixed point p and keeping at a distance r.
with C(r) the distance walked
Delevopable surfaces: surfaces whose Gaussian curvature is 0 everywhere
κG=limr→0
(2π r−C (r ))⋅ 3π r3
Gaussian Curvature
κG>0 κG<0
Mean Curvature
Divergence of the surface normal divergence is an operator that measures a vector
field's tendency to originate from or converge upon a given point
Minimal surface and minimal area surfaces A surface is minimal when its mean curvature is 0
everywhere All minimal area surfaces have mean curvature 0
The surface tension of an interface, like a soap bubble, is proportionalto its mean curvature
Mean Curvature
Let A be the area of a disk around p. The mean curvature
the mean derivative is (twice) the divergence of the normal
2 κ̄=∇⋅n= ∂∂ xn+ ∂
∂ yn+ ∂
∂ zn
2 n= limdiamA0
∇ AA
Gaussian curvature on a triangle mesh
It's the angle defect over the area
Gauss-Bonnet Theorem: The integral of the Gaussian Curvature on a closed surface depends on the Euler number
κG(v i)=1
3A(2 π− ∑
t j adj vi
θ j)
∫S
κG=2π χ
Example data structure
SimplestList of triangles:
For each triangle store its coords.
1. (3,-2,5), (3,6,2), (-6,2,4)2. (2,2,4), (0,-1,-2), (9,4,0)3. (1,2,-2), (3,6,2), (-4,-5,1) 4. (-8,2,7), (-2,3,9), (1,2,-2)
How to find any adjacency?Does it store FV?
Example data structure
Slightly betterList of unique vertices with indexed faces
Storing the FV relation
Faces: 1. 1 2 4 2. 5 7 6 3. 1 5 2 4. 3 4 7 5. 1 7 5
Vertices: 1. (-1.0, -1.0, -1.0) 2. (-1.0, -1.0, 1.0) 3. (-1.0, 1.0, -1.0) 4. (-1, 1, 1.0) 5. (1.0, -1.0, -1.0) 6. (1.0, -1.0, 1.0) 7. (1.0, 1.0, -1.0) 8. (1.0, 1.0, 1.0)
Example data structure
Issue of AdjacencyVertex, Edge, and Face Structures
Each element has list of pointers to all incident elements
Queries depend only on local complexity of mesh!Slow! Big! Too much work to maintain!Data structures do not have fixed size
V E F
Example data structure
Winged edge Classical real smart structureNice for generic polygonal meshesUsed in many sw packages
V E F
Winged Edge
Winged edge CompactAll the query requires some kind of
''traversal''Not fitted for rendering...
4
Representing Real World Surfaces
Analytic definition falls short of representing real world surfaces in a tractable way
... surfaces can be represented by cell complexes
5
Cell complexes (meshes)
In tu it ive descr ip tion: a con tinuous surface d iv ided in po lygons
quadrilaterals (quads)
triangles Generic polygons
6
Ce ll Com p lexes (m eshes)In nature, meshes arise in a variety of
contexts:Cells in organic tissuesCrystalsMolecules…Mostly convex but irregular cellsCommon concept: complex shapes can be
described as collections of simple building blocks
6
6
7
Cell Complexes (meshes)
m ore form al defin ition a ce ll is a convex po lytope in a proper face of a ce ll is a convex po lytope
in
8
Cell Complexes (meshes)
a co llection of ce lls is a com plex iff every face o f a ce ll be longs to the com plex For every ce lls C and C ’, the ir in tersection
e ither is em pty or is a com m on face of both
9
Maximal Cell Complex
the order of a cell is the number of its sides (or vertices)
a complex is a k-complex if the maximum of the order of its cells is k
a cell is maximal if it is not a face of another cell a k-complex is maximal iff all maximal cells have
order k short form : no dangling edges!
10
Simplicial Complex
A ce ll com plex is a sim plicial com plex when the ce lls are s im plexes
A d-s im p lex is the convex hu ll o f d+1 po ints in
0-simplex 1-simplex 2-simplex 3-simplex
11
Meshes, at last
When talking of triangle mesh the intended meaning is a maximal 2-simplicial complex
Topology vs Geometry
Di un complesso simpliciale e' buona norma distinguereRealizzazione geometrica
Dove stanno effettivamente nello spazio i vertici del nostro complesso simpliciale
Caratterizzazione topologicaCome sono connessi combinatoriamente i vari
elementi
Topology vs geometry 2
Nota: Di uno stesso oggetto e' possibile dare rappresentazioni con eguale realizzazione geometrica ma differente caratterizzazione topologica (molto differente!) Demo kleine
Nota: Di un oggetto si puo' dire molte cose considerandone solo la componente topologicaOrientabilitacomponenti connessebordi
14
Manifoldness
a surface S is 2-manifold iff:the neighborhood of each point is
homeomorphic to Euclidean space in two dimensionor … in other words..
the neighborhood of each point is homeomorphic to a disk (or a semidisk if the surface has boundary)
15
Orientability
A surface is orientable if it is possible to make a consistent choice for the normal vector …it has two sides
Moebius strips, klein bottles, and non manifold surfaces are not orientable
Incidenza Adiacenza
Due simplessi σ e σ' sono incidenti se σ è una faccia propria di σ' o vale il viceversa.
Due k-simplessi sono m-adiacenti (k>m) se esiste un m-simplesso che è una faccia propria di entrambi.Due triangoli che condividono
un edge sono 1 -adiacentiDue triangoli che condividono
un vertice sono 0-adiacenti
Relazioni di Adiacenza
Per semplicità nel caso di mesh si una relazione di adiacenza con un una coppia (ordinata!) di lettere che indicano le entità coinvolteFF adiacenza tra triangoliFV i vertici che compongono un triangoloVF i triangoli incidenti su un dato vertice
Relazioni di adiacenza
Di tutte le possibili relazioni di adiacenza di solito vale la pena se ne considera solo un sottoinsieme (su 9) e ricavare le altre proceduralmente
Relazioni di adiacenza FF ~ 1-adiacenza
EE ~ 0 adiacenza
FE ~ sottofacce proprie di F con dim 1
FV ~ sottofacce proprie di F con dim 0
EV ~ sottofacce proprie di E con dim 0
VF ~ F in Σ : V sub faccia di F
VE ~ E in Σ : V sub faccia di E
EF ~ F in Σ : E sub faccia di F
VV ~ V' in Σ : Esiste E (V,V')
Partial adiacency
Per risparmiare a volte si mantiene una informazione di adiacenza parziale VF* memorizzo solo un riferimento dal
vertice ad una delle facce e poi 'navigo' sulla mesh usando la FF per trovare le altre facce incidenti su V
Relazioni di adiacenza
In un 2-complesso simpliciale immerso in R3, che sia 2 manifoldFV FE FF EF EV sono di cardinalità bounded
(costante nel caso non abbia bordi)|FV|= 3 |EV| = 2 |FE| = 3 |FF| <= 2 |EF| <= 2
VV VE VF EE sono di card. variabile ma in stimabile in media|VV|~|VE|~|VF|~6|EE|~10F ~ 2V
22
Genus
The Genus of a closed surface, orientable and 2-manifold is the maximum number of cuts we can make along non intersecting closed curves without splitting the surface in two.
…also known as the number of handles
0 1 2
23
Euler characteristic
24
Euler characteristics
= 2 for any simply connected polyhedron proof by construction… play with examples:
χ
25
Euler characteristics
let’s try a more complex figure…
why =0 ?
26
Euler characteristics
where g is the genus of the surface
χ=2−2g
27
Euler characteristics
let’s try a more complex figure…remove a face. The surface is not closed anymore
why =-1 ?
28
Euler characteristics
where b is the num ber of borders of the surface
χ=2−2g−b
29
Euler characteristics
Remove the border by adding a new vertex and connecting all the k vertices on the border to it.
A A’
X' = X + V' -E' + F' = X + 1 – k + k = X +1
Differential quantities:normals
The (unit) normal to a point is the (unit) vector perpendicular to the tangent plane
implicit surface : f ( x , y , z )=0
n= ∇ f[∇ f ]
parametric surface :f (u ,v)=( f x (u ,v) , f y (u ,v) , f z (u ,v))
n= ∂∂u
f × ∂∂ v
f
Normals on triangle meshes
Computed per-vertex and interpolated over the faces
Common: consider the tangent plane as the average among the planes containing all the faces incident on the vertex
Normals on triangle meshes
Does it work? Yes, for a “good” tessellationSmall triangles may change the result
dramatically Weighting by edge length / area / angle
helps
Differential quantities: Curvature
The curvature is a measure of how much a line is curve
x(t) , y (t) a parametric curveϕ tangential angle%s arc lenght
κ≡d ϕds
=d ϕ/dtds /dt
=d ϕ/dt
√( dxdt
)2
+( dydt
)2=
d ϕ/dt
√ x ' 2+ y ' 2
κ= x ' y ' '− y ' x ' '( x ' 2+ y ' 2)3/2
Curvature on a surface
Given the normal at point p and a tangent direction
The curvature along is the 2D curvature of the intersection between the plane and the surface
θθ
Curvatures
A curvature for each direction Take the two directions for which curvature is max and
min
the directions of max and min curvature are orthogonal
κ1 ,κ2 principal curvaturese1 , e2 principal directions
[Meyer02]
Gaussian and Mean curvature
Gaussian curvature: the product of principal curvatures
Mean curvature: the average of principal curvatures
κG≡Κ≡κ1⋅κ2
κ̄≡H≡κ1+κ2
2
Examples..
Red:low → red:high (not in the same scale)
mean gaussian min max
[Meyer02]
Gaussian Curvature
Gaussian curvature is an intrinsic propertyIt can be computed by a bidimensional
inhabitant of the surface by walking around a fixed point p and keeping at a distance r.
with C(r) the distance walked
Delevopable surfaces: surfaces whose Gaussian curvature is 0 everywhere
κG=limr→0
(2π r−C (r ))⋅ 3π r3
Gaussian Curvature
κG>0 κG<0
Mean Curvature
Divergence of the surface normal divergence is an operator that measures a vector
field's tendency to originate from or converge upon a given point
Minimal surface and minimal area surfaces A surface is minimal when its mean curvature is 0
everywhere All minimal area surfaces have mean curvature 0
The surface tension of an interface, like a soap bubble, is proportionalto its mean curvature
Mean Curvature
Let A be the area of a disk around p. The mean curvature
the mean derivative is (twice) the divergence of the normal
2 κ̄=∇⋅n= ∂∂ xn+ ∂
∂ yn+ ∂
∂ zn
2 n= limdiamA0
∇ AA
Gaussian curvature on a triangle mesh
It's the angle defect over the area
Gauss-Bonnet Theorem: The integral of the Gaussian Curvature on a closed surface depends on the Euler number
κG(v i)=1
3A(2 π− ∑
t j adj vi
θ j)
∫S
κG=2π χ
Example data structure
SimplestList of triangles:
For each triangle store its coords.
1. (3,-2,5), (3,6,2), (-6,2,4)2. (2,2,4), (0,-1,-2), (9,4,0)3. (1,2,-2), (3,6,2), (-4,-5,1) 4. (-8,2,7), (-2,3,9), (1,2,-2)
How to find any adjacency?Does it store FV?
Example data structure
Slightly betterList of unique vertices with indexed faces
Storing the FV relation
Faces: 1. 1 2 4 2. 5 7 6 3. 1 5 2 4. 3 4 7 5. 1 7 5
Vertices: 1. (-1.0, -1.0, -1.0) 2. (-1.0, -1.0, 1.0) 3. (-1.0, 1.0, -1.0) 4. (-1, 1, 1.0) 5. (1.0, -1.0, -1.0) 6. (1.0, -1.0, 1.0) 7. (1.0, 1.0, -1.0) 8. (1.0, 1.0, 1.0)
Example data structure
Issue of AdjacencyVertex, Edge, and Face Structures
Each element has list of pointers to all incident elements
Queries depend only on local complexity of mesh!Slow! Big! Too much work to maintain!Data structures do not have fixed size
V E F
Example data structure
Winged edge Classical real smart structureNice for generic polygonal meshesUsed in many sw packages
V E F
Winged Edge
Winged edge CompactAll the query requires some kind of
''traversal''Not fitted for rendering...